The space \(\mathcal {A}_F^\mathfrak {g}\)  parametrizes connections \(A \,\in \, \Omega ^1 (\Sigma ,\, \mathfrak {g})\)  with curvature \(F_A \,=\, 0 \)  and
\(A \)  having the form \(\lambda ds \)  on a neighbourhood of
the boundary (where \(s \,\in \, [0,\, 2 \pi ]\)  is a coordinate on the boundary and
\(\lambda \)  is a constant in \(\mathfrak {g}\) ).
Then \(\mathcal {M}_g(\Sigma )\)  consists of the quotient of \(A^\mathfrak {g}_F\) 
by the group of gauge transformations equal to the identity on a
neighbourhood of the boundary.
The space \(\mathcal {M}_g(\Sigma )\)  is homeomorphic
to \({\mathcal {N}}^\mathfrak {g}\)  (see [1]}).
